BES 026

Dynamics of Rigid Bodies

Students’ Activity Sheet Module # 6
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

L E S S O N T I T L E : C U RV I L I N E A R M O T I O N : Materials:
NORMAL AND TANGENTIAL COMPONENTS Textbook, Notebook, Pen, Calculator

Lesson Objectives: References:

Main Textbook: Engineering Mechanics: Statics and
At the end of this module, you should be able to: Dynamics by R.C. Hibbeler (https://drive.google.com/
1. Acknowledge the normal and tangential file/d/1LSZHHta3BIh7GugVLaTlyz4oPUGvFqKi/
components of velocity and acceleration in view?usp=sharing)
curvilinear motion. You can copy the links and open them on Google to
obtain a softcopy of the file. You can place it on your
phone for easy access wherever you are 😄

Productivity Tip:

IF YOU ARE NOT PATIENT ENOUGH IN

MAKING MESSES, YOU CANNOT MAKE
MASTERPIECES. - Unknown
So don’t worry about committing mistakes! You’ll learn
more from them than from victories! Always crave for
knowledge and don’t be afraid of failing. Try and try and
try and try and try until Dynamics feels like 1+0! (Oh diba,
mas madali pa sa 1+1). Take note, you’re a beginner
today and tomorrow, you’re a champion!

A. LESSON PREVIEW
Introduction
Now, it’s time for another topic! This is somehow related to Projectile Motion but this is more general and we
call this Curvilinear Motion. Projectile Motion is actually Curvilinear Motion. Curvilinear Motion differs from
Rectilinear Motion such that the latter involves curve paths, not straight lines. So we are going to simplify our
study of Curvilinear Motion by describing this motion using two axes - the n and t coordinate axes, which act
normal (n) and tangent (t) to the path respectively, at that instant considered. Now, let’s get to it. Turn to the
next page! 😀

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 1 What I Know Chart (PART 1)

Instruction: Before proceeding to the main lesson, write three things you know or have learned from the
previous lesson in the first column “What I Know”. The third column “What I Learned” will be filled out at the
end of the module.

WHAT I KNOW QUESTIONS WHAT I LEARNED (Activity 4)

The velocity of a particle in a

1 1 curved path always acts
_______ to the path.

How do you get the slope of a

2 2
curve?

Write the equation of the radius

3 3
of curvature. No looking!

B. MAIN LESSON
ACTIVITY 2 Content Notes
In this next activity, you will be presented with presentation slides and a detailed explanation of the content of
each slide. You are to write down the key points of the lecture that are critical to your evaluation. Make sure
that your pen and paper are beside you so you can go with the flow of the lesson. Enjoy reading and have fun
learning 😍 See next page for the main lesson!

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 BES 026
Dynamics of Rigid Bodies
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

A particle is moving along a curved path as shown. At this instant, it has a radius of curvature r with
respect to O’. This radius of curvature changes unless the path is a circle.
At this instant, the velocity always acts tangent to the path.
So we are going to denote it with a unit vector ut (this is like a unit). And this italicized v is equal to the rate
of change of position (ds/dt).

As for the acceleration, it has two components - the normal and tangential. The normal component of
acceleration acts towards the radius of curvature (towards O’) while the tangential compoentn of
acceleration acts tangent to the path.
Again, we denote it with a unit vector ut and un. at is the rate of change of velocity (dv/dt) and the formula of
an is given below.

The magnitude of the the acceleration is given by:

The radius of curvature of any path (that has an equation of

y=f(x) is given by:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 1

When the skier reaches point A along the parabolic path as shown below, he has a speed of 6 m/s which is
increasing at 2 m/s2. Determine the direction of his velocity and the direction and magnitude of his acceleration
at this instant. Neglect the size of the skier in the calculation.

Solution:
To get the direction of his velocity (angle), we take a look at the equation and then equate the derivative to
tan θ. So we get the derivative of the curve which is dy/dx = x/10, and at x=10, dy/dx = 1 so we equate 1 to
tan θ and that is equal to 45o.

Next is the direction and magnitude of his acceleration. So we first get the radius of curvature since we’ll be
needing the normal acceleration.

The tangential acceleration is given which is 2 m/s2.

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As seen in the figure, a sketch of the normal and tangential acceleration is drawn and the resultant of these
two is the total acceleration.

The direction is always of course with respect to the horizontal.

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Dynamics of Rigid Bodies
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Name: __________________________________
Section:_________________________________ Class Number: _______________
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SAMPLE PROBLEM NO. 2

A race car C travels around the horizontal circular track that has a radius of 300 ft. If the car increases its
speed at a constant rate of 7 ft/s2, starting from rest, determine the time needed for it to reach an acceleration
of 8 ft/s2. What is its speed at this instant?

At constant acceleration, the velocity can be obtained by using the equations of kinematics with constant
acceleration. Since at = 7 ft/s2, an = v2/r. The velocity ay this instant is:

The velocity is calculated as a function of time since the time is still unknown.

The time needed is:

So the velocity at this instant is equal to:

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 3

The boxes travel along the industrial conveyor. If a box starts from rest at A and increases its speed such that
at = (0.2t) m/s2, where t is in seconds, determine the magnitude of its acceleration when it arrives at point B.

Solution:
We can get the position of the box when it gets to B with respect to the origin A. That is 3 + (2px2)/4 =
6.142 m. Since we only have the acceleration equation, we integrate twice to get the position equation.

Now that we have the time, we can now go back to the equations of acceleration and velocity.

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
Home Module
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 3 Skill-Building Activities

Now, you will be presented with a brief activity that contains both objective and problem solving type questions.
You can check your answers against the Key to Corrections found at the end of this module. But as much as
possible, don’t peek if you still haven’t tried answering the questions 😅 Write your score on the space
provided. Good Luck! Have fun answering!

WHAT’S
Activity 3 Questions YOUR
SCORE?

When x = 10 ft, the crate has a speed of 20 ft/s which is increasing at 6 ft/s2. Determine the direction of
the crate’s velocity and the magnitude of the crate’s acceleration at this instant.

ACTIVITY 4 What I Know Chart (PART 2)

Now, go back to Activity 1 and answer the questions filling out the “What I Know Column” after going through
the main lesson and answering the questions from Activity 3.

ACTIVITY 5 Check For Understanding

Now, I’m sure that you have learned enough and are ready to take a short quiz. Of course, the questions that
will be given are more challenging than Activity 3. The schedule will be announced in class. Instructions:
You will write your solutions and answers in a short coupon bond. You may use more than one paper. But only
a limited number will be submitted. (Note: For problem solving type: for values less than one, round off to up to
four significant figures {e.g. 0.1234, 0.5780, 0.0069 are values with four significant figures}, and for values
greater than one, round off to three decimal places {e.g. 3.178, 222.900, 93.232 are values with three decimal
places}).

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 6
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

C. LESSON WRAP-UP
ACTIVITY 6 Thinking About Learning

You are done with the session! Let’s track your progress!
Period 1 Period 2 Period 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ACTIVITY 6.1 Unanswered Questions

In this activity, I’m asking you to write your questions that are still unanswered at this time of reading. If there’s
any thing that still confuses you, don’t hesitate to write them down below. Make sure though that these
questions are not included in the FAQs section written below/ on the next page. So I suggest you to read the
FAQs first before proceeding to this activity 😊
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________

FREQUENTLY ASKED QUESTIONS (FAQs)

QUESTION How will we know when to integrate or not?

ANSWER Most of the time, if there is constant acceleration, it will be indicated in the problem. If you see
that the equations are functions of time, then you need to integrate.
QUESTION Do we have to place ut and un all the time?

ANSWER Yes because if you don’t, you might get confused and end up adding them in the end.

KEY TO CORRECTIONS
{*Teacher lists answers to Activities #3 and #5.
**Give Answer Key for Quizzes in Teachers’ Guide only.
***This can be a rubric for open-ended activities.}
See Answer Key from ACTIVITY 3 Skill-Building Activities
1. Θ = 39.8 degrees CCW with respect to the horizontal (20 points), a = 16.3 ft/s2 (50 points).

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

LESSON TITLE: ADDITIONAL PROBLEMS IN Materials:

CURVILINEAR MOTION AND PROBLEM SET Textbook, Notebook, Pen, Calculator

Lesson Objectives: References:

Main Textbook: Engineering Mechanics: Statics and
At the end of this module, you should be able to: Dynamics by R.C. Hibbeler (https://drive.google.com/
1. Generate your own problems regarding file/d/1LSZHHta3BIh7GugVLaTlyz4oPUGvFqKi/
curvilinear motion that involves normal and view?usp=sharing)
tangential components. You can copy the links and open them on Google to
2. Begin drafting your problem set in preparation for obtain a softcopy of the file. You can place it on your
your next quiz. phone for easy access wherever you are 😄

Productivity Tip:

ABILITY TO FIND THE ANSWERS IS MORE

IMPORTANT THAN THE ABILITY TO KNOW
THE ANSWERS. - Amit Kalantri
Subjects like this doesn’t require a lot of memorization. It
requires knowledge of the basic concepts and applying
them to problems to find the answers. It’s not enough
that you memorize a formula. You have to understand
where it came from and where to use them. If you master
it, you can solve any problem in Mechanics! 😀

A. LESSON PREVIEW
Introduction
In this module, I will present more problems from Curvilinear Motion because there are just too many of them!
Make sure that you have grasped the concepts of Normal and Tangential Components. Just to give you a
recall, a particle moving along a curved path can be described by its position, and the velocity and acceleration
in normal and tangential components. The velocity always acts tangent to the path and the acceleration has
both normal and tangential components. The total acceleration can be obtained by getting the resultant of the
two components. Furthermore, the direction can be solved by basic trigonometric equations. So, let’s get to the
next page! 😀

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 1 What I Know Chart (PART 1)

Instruction: Before proceeding to the main lesson, write three things you know or have learned from the
previous lesson in the first column “What I Know”. The third column “What I Learned” will be filled out at the
end of the module.

WHAT I KNOW QUESTIONS WHAT I LEARNED (Activity 4)

If acceleration is given as a
function of position, what
1 1
equation are you going to use
to analyze motion?
Suppose that a car moves
along a straight path, what do
2 2
you think is the radius of
curvature of that path?

What type of path has a

3 3
constant radius of curvature?

B. MAIN LESSON
ACTIVITY 2 Content Notes
In this next activity, you will be presented with presentation slides and a detailed explanation of the content of
each slide. You are to write down the key points of the lecture that are critical to your evaluation. Make sure
that your pen and paper are beside you so you can go with the flow of the lesson. Enjoy reading and have fun
learning 😍 See next page for the main lesson!

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
Home Module
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 1

The boat is traveling along a circular path with a speed of v = (0.0625t2) m/s, where t is in seconds. Determine
the magnitude of its acceleration when t = 10 s.

Solution:

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 2

If the car decelerates uniformly along the curved road from 25 m/s at A to 15 m/s at C, determine the
acceleration of the car at B.

Solution:

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Dynamics of Rigid Bodies
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SAMPLE PROBLEM NO. 3

If the motorcycle has a deceleration of a = -(0.001s) m/s2 and its speed at position A is 25 m/s, determine the
magnitude of its acceleration when it passes point B.

Solution:
Well, at this time, you can see that acceleration is a function of s. We have to remember the last equation
of kinematics which is ads=vdv, and integrate from there.

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 4

The car travels up a hill with a speed of v = (0.2s) m/s, where s is in meters, measured from A. Determine the
magnitude of its acceleration when it is at point s = 50m, where r = 500m.

Solution:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 5

The car is traveling along the road with a speed of v = (2s) m/s, where s is in meters. Determine the magnitude
of its acceleration when s = 10 m.

Solution:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 3 Skill-Building Activities

Now, you will be presented with a problem set that you will be submitting on our next face-to-face meeting. The
format is as follows:

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Dynamics of Rigid Bodies
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Activity 3 Problem Set Questions

If the car passes point A with a speed of 20 m/s and begins to increase its speed at a constant rate of at =
0.5 m/s2, determine the magnitude of the car’s acceleration when s = 101.68 m and x = 0.

The box of negligible size is sliding down along a curved path defined by the parabola y = 0.4x2. When it
is at A (xA = 2 m, yA = 1.6 m), the speed is v = 8 m/s and the increase in speed is dv/dt = 4 m/s2.
Determine the magnitude of the acceleration of the box at this instant.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________
The motorcycle is traveling at a constant speed of 60 km/h. Determine the magnitude of its acceleration
when it is at point A.

The race car has an initial speed vA = 15 m/s at A. If it increases its speed along the circular track at the
rate at = (0.4s) m/s2, where s is in meters, determine the time needed for the car to travel 20 m. Take r =
150 m.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 4 What I Know Chart (PART 2)

Now, go back to Activity 1 and answer the questions filling out the “What I Know Column” after going through
the main lesson and answering the questions from Activity 3.

ACTIVITY 5 Check For Understanding

Now, I’m sure that you have learned enough and are ready to take a short quiz. Of course, the questions that
will be given are more challenging than Activity 3. The schedule will be announced in class. Instructions:
You will write your solutions and answers in a short coupon bond. You may use more than one paper. But only
a limited number will be submitted. (Note: For problem solving type: for values less than one, round off to up to
four significant figures {e.g. 0.1234, 0.5780, 0.0069 are values with four significant figures}, and for values
greater than one, round off to three decimal places {e.g. 3.178, 222.900, 93.232 are values with three decimal
places}).

C. LESSON WRAP-UP
ACTIVITY 6 Thinking About Learning

You are done with the session! Let’s track your progress!
Period 1 Period 2 Period 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ACTIVITY 6.1 Unanswered Questions

In this activity, I’m asking you to write your questions that are still unanswered at this time of reading. If there’s
any thing that still confuses you, don’t hesitate to write them down below. Make sure though that these
questions are not included in the FAQs section written below/ on the next page. So I suggest you to read the
FAQs first before proceeding to this activity 😊
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 7
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

FREQUENTLY ASKED QUESTION (FAQ)

QUESTION What is dv/ds?

ANSWER You might notice that in the equation ads=vdv, ds is usually cross multiplied to get dv/ds. dv/ds
is simply the derivative of velocity with respect to position so if you have the equation of
velocity as a function of position (v=f(s)) then dv/ds is simply the derivative of the equation.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

LESSON TITLE: NEWTON’S SECOND LAW OF Materials:

MOTION Textbook, Notebook, Pen, Calculator

Lesson Objectives: References:

Main Textbook: Engineering Mechanics: Statics and
At the end of this module, you should be able to: Dynamics by R.C. Hibbeler (https://drive.google.com/
1. State Newton’s Second Law of Motion and to file/d/1LSZHHta3BIh7GugVLaTlyz4oPUGvFqKi/
define mass and weight. view?usp=sharing)
2. Analyze the accelerated motion of a particle using You can copy the links and open them on Google to
the equation of motion with different coordinate obtain a softcopy of the file. You can place it on your
systems. phone for easy access wherever you are 😄

Productivity Tip:

IF EVERYBODY WERE LIKE EVERYBODY

ELSE, HOW BORING IT WOULD BE. THE
THINGS THAT MAKE ME DIFFERENT ARE
THE THINGS THAT MAKE ME, ME! - Piglet and
Eeyore, Winnie the Pooh
This, I always tell myself, do not try to be the best, try to
be the only! Be the person who solved 20 problems even
when the teacher told you to solve only 10. Be the
person who read an entire chapter even when the
teacher told you to focus on only three pages. Be more!
You can do this. You can always be more than what you
were yesterday!

A. LESSON PREVIEW
Introduction
We’re finally done with Kinematics. Now, it’s time that we discuss Kinetics. Kinetics is a branch of dynamics
that deals with the relationship between the change in motion of a body and the forces that cause this change.
The basis for kinetics is Newton’s second law, which states that when an unbalanced force acts on a particle,
the particle will accelerate in the direction of the force with a magnitude that is proportional to the force. This
law can be verified experimentally by applying a known unbalanced force F to a particle, and then measuring
the acceleration a. Since the force and acceleration are directly proportional, the constant of proportionality, m,
may be determined from the ratio m = F/a. This positive scalar m is called the mass of the particle. Being
constant during any acceleration, m provides a quantitative measure of the resistance of the particle to a
change in its velocity, that is its inertia. Now that’s the main concept. Turn to the next page to see the equation
we will be using! 😎

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 1 What I Know Chart (PART 1)

Instruction: Before proceeding to the main lesson, write three things you know or have learned from the
previous lesson in the first column “What I Know”. The third column “What I Learned” will be filled out at the
end of the module.

WHAT I KNOW QUESTIONS WHAT I LEARNED (Activity 4)

State Newton’s Second Law of

1 1
Motion.

What is the formula of frictional

2 2
force?

The Normal Force acts

3 3
_______ to the surface.

B. MAIN LESSON
ACTIVITY 2 Content Notes
In this next activity, you will be presented with presentation slides and a detailed explanation of the content of
each slide. You are to write down the key points of the lecture that are critical to your evaluation. Make sure
that your pen and paper are beside you so you can go with the flow of the lesson. Enjoy reading and have fun
learning 😍 See next page for the main lesson!

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

If the mass of particle is m, Newton’s Second Law of Motion may be written in mathematical form as:

The above equation, which is referred to as the equation of motion, is one of the most important
formulations in mechanics.

THE EQUATION OF MOTION

When more than one force acts on a particle, the resultant force is determined by a vector summation of all
the forces or in general:

FR or the resultant force of a set of forces is equated to the product of the mass of the object and its
corresponding acceleration.

THE EQUATION OF MOTION: RECTANGULAR COORDINATES

When a particle moves relative to an inertial x, y, z frame of reference, the forces acting on the particle, as
well as its acceleration, can be expressed in terms of their i, j, k components. Applying the equation of
motion, we have:

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Name: __________________________________
Section:_________________________________ Class Number: _______________
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SAMPLE PROBLEM NO. 1

The 50-kg crate shown below rests on a horizontal surface for which the coefficient of kinetic friction is mk =
0.3. If the crate is subjected to a 400-N towing force as shown, determine the velocity of the crate in 3 s
starting from rest.

Solution:
It’s always very important to draw the Free Body Diagram first before using the equation.
So as you can see, we have four forces acting:
First is the weight of the crate which should
obviously be directed downwards. Converting 50
kilos to N, we simply multiply it with g (the
gravitational acceleration which is 9.81 m/s2 if
we’re using SI and 32.2 ft/s2 if we’re using
English). So since we’re using SI, we multiply 50
by 9.81 which is going to give us 490.5 Newtons.
Second is of course the force P = 400 N, and then
the Normal force which is always perpendicular to
the surface and then lastly is the Frictional force
which is always opposite the direction of motion.
Frictional force is equal to the coefficient of kinetic
friction multiplied by the normal force. So in short,
F = mkN. So if a is to the right, F should be to
the left and vice versa. The direction of
acceleration is just assumed.
So there we have it, all four forces. Now, it’s time
that we use the Equation of Motion.
Since our diagram is only two-dimensional:

There will be no acceleration in the y axis since

the crate won’t go up due to the force P.

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Solving the two equations simultaneously, we have:

The velocity is then obtained by using the equation of kinematics:

SAMPLE PROBLEM NO. 2

A 10-kg projectile is fired vertically upward from the ground, with an
initial velocity of 50 m/s. Determine the maximum height to which it
will travel.

Solution:

The free body diagram is shown above.

The result indicates that the projectile, like every other object
having free flight motion near the Earth’s surface is subjected to a constant downward acceleration of 9.81
m/s2.
Using the equation of kinematics since there is constant acceleration:

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 3

The 10-kg block is subjected to the forces shown. In each case,determine its velocity at s = 8 m if v = 3 m/s at
s = 0. Motion occurs to the right.

Solution:
Again, the free body diagram is sketched first. Then Newton’s Second Law of Motion was used and then
the velocity was obtained from using the equations of kinematics with constant acceleration.

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

For the second item, there’s a need for integration because the force is a function of position.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 3 Skill-Building Activities

Now, you will be presented with a brief activity that contains both objective and problem solving type questions.
You can check your answers against the Key to Corrections found at the end of this module. But as much as
possible, don’t peek if you still haven’t tried answering the questions 😅 Write your score on the space
provided. Good Luck! Have fun answering!

WHAT’S
Activity 3 Questions YOUR
SCORE?

The 10-kg block is subjected to the forces shown. In each case, determine its velocity when t = 2 s if v =
1
0 when t = 0.

ACTIVITY 4 What I Know Chart (PART 2)

Now, go back to Activity 1 and answer the questions filling out the “What I Know Column” after going through
the main lesson and answering the questions from Activity 3.

ACTIVITY 5 Check For Understanding

Now, I’m sure that you have learned enough and are ready to take a short quiz. Of course, the questions that
will be given are more challenging than Activity 3. The schedule will be announced in class. Instructions:
You will write your solutions and answers in a short coupon bond. You may use more than one paper. But only
a limited number will be submitted. (Note: For problem solving type: for values less than one, round off to up to
four significant figures {e.g. 0.1234, 0.5780, 0.0069 are values with four significant figures}, and for values
greater than one, round off to three decimal places {e.g. 3.178, 222.900, 93.232 are values with three decimal
places}).

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 8
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

C. LESSON WRAP-UP
ACTIVITY 6 Thinking About Learning

You are done with the session! Let’s track your progress!
Period 1 Period 2 Period 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ACTIVITY 6.1 Unanswered Questions

In this activity, I’m asking you to write your questions that are still unanswered at this time of reading. If there’s
any thing that still confuses you, don’t hesitate to write them down below. Make sure though that these
questions are not included in the FAQs section written below/ on the next page. So I suggest you to read the
FAQs first before proceeding to this activity 😊
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________

FREQUENTLY ASKED QUESTIONS (FAQs)

QUESTION How do we know the direction of acceleration?

ANSWER Always assume the direction of acceleration if not indicated. In the end, your assumption is
correct when it is positive. If the value if negative, that just means that the direction of
acceleration is opposite of your assumption.
QUESTION When to integrate the motion equation?

ANSWER Always take note that whenever there is a function of something, integration is mandatory.

KEY TO CORRECTIONS
{*Teacher lists answers to Activities #3 and #5.
**Give Answer Key for Quizzes in Teachers’ Guide only.
***This can be a rubric for open-ended activities.}
See Answer Key from ACTIVITY 3 Skill-Building Activities
1. For a) v=20 m/s (20 points), For b) v=4 m/s (20 points).

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

LESSON TITLE: ADDITIONAL PROBLEMS IN Materials:

NEWTON’S SECOND LAW OF MOTION AND Textbook, Notebook, Pen, Calculator
PROBLEM SET
References:
Lesson Objectives: Main Textbook: Engineering Mechanics: Statics and
Dynamics by R.C. Hibbeler (https://drive.google.com/
At the end of this module, you should be able to: file/d/1LSZHHta3BIh7GugVLaTlyz4oPUGvFqKi/
1. Generate your own problems regarding Newton’s view?usp=sharing)
Second Law of Motion. You can copy the links and open them on Google to
2. Begin drafting your problem set in preparation for obtain a softcopy of the file. You can place it on your
your next quiz. phone for easy access wherever you are 😄

Productivity Tip:

IF YOU’RE GOING THROUGH HELL, WHY

STOP AT HELL. - Unknown
This couldn’t have been said any better. You might be
going through something so difficult now but all it takes
is your courage and determination to finish the fight. You
have a goal! You are going to be an engineer. But this is
only possible when you get up there and move! 😀

A. LESSON PREVIEW
Introduction
Now I present to you more problems about Newton’s Second Law of Motion. In the last module, you knew that
the resultant force of a set of forces is equal to the product of mass and acceleration. In mathematical
equation, EF = ma. In dealing with diverse directions of motion, the forces along the x-axis will correspond to
the acceleration along the x-axis as well. We discussed motion in the horizontal and vertical direction last time.
Today, we will analyze some inclined surfaces. Take note that in a given surface, the weight, normal force, and
the frictional force (which is a function of the normal force and the coefficient of kinetic friction) are always
present alongside the unbalanced force that causes the motion. Now, let’s get to it. Turn to the next page! 😀

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 1 What I Know Chart (PART 1)

{*Teachers write down 2-3 questions on the second column about the topic for the day and ask students to
write on the first column what they know. The third column is left blank at this time.
**Teachers’ questions can relate to information, processes, hypotheses, opinions, feelings, procedures,
reasoning, solutions, and others.)
Instruction: Before proceeding to the main lesson, write three things you know or have learned from the
previous lesson in the first column “What I Know”. The third column “What I Learned” will be filled out at the
end of the module.

WHAT I KNOW QUESTIONS WHAT I LEARNED (Activity 4)

What is the coefficient of kinetic

1 1
friction in a smooth surface?

How many kilograms are in a

2 2
Megagram?

When cutting and exposing

3 3 internal forces in ropes, are
they compressive in nature?

B. MAIN LESSON
ACTIVITY 2 Content Notes
{*Pre-printed and complete content which can be articles, excerpts, lecture notes, a case study, a situationer, a
set of procedures, data, graphs.
**Include instructions for note-taking, highlighting, or outlining.
***Vary the presentation of content to sustain student attention and interest. Maximize use of graphic
organizers and illustrations to aid comprehension.}
In this next activity, you will be presented with presentation slides and a detailed explanation of the content of
each slide. You are to write down the key points of the lecture that are critical to your evaluation. Make sure
that your pen and paper are beside you so you can go with the flow of the lesson. Enjoy reading and have fun
learning 😍 See next page for the main lesson!

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 1

Write the equations of motion in the x and y directions for the 10-kg block.

Solution:
To simply the solutions involving inclined motion, change the orientation of x and y. Make the x-axis parallel
to the surface and the y-axis perpendicular to the surface.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 2

The motor winds in the cable with a constant acceleration, such that the 20-kg crate moves a distance s = 6 m
in 3 s, starting from rest. Determine the tension developed in the cable. The coefficient of kinetic friction
between the crate and the plane is mk = 0.3.

Solution:
Challenge: Draw the FBD of this system then to expose the tension cut the rope and isolate the crate.
Note: Tension always acts away from the cut section.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 3

The 2-Mg (Megagram) car is being towed by a winch. If the winch exerts a force of T = 100(s + 1) N on the
cable, where s is the displacement of the car in meters, determine the speed of the car when s = 10 m, starting
from rest. Neglect rolling resistance of the car.

Solution:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 3 Skill-Building Activities

{*Teacher designs tasks around the content that develop learning, build skills in comprehension, critical
thinking, creativity, and communication, practice technical skills, and foster value formation.
**There can be more than one activity. Ex. First activity can be organizing information from the content into a
chart and the second activity can be answering the guide questions.
***Teacher adds this prompt for self-assessment: “Check your answers against the Key to Corrections found at
the end of this SAS. Write your score on your paper.”}
Now, you will be presented with a problem set that you will be submitting on our next face-to-face meeting. The
format is as follows:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

Activity 3 Problem Set Questions

Determine the acceleration of the blocks when the system is released. The coefficient of kinetic friction is
mk, and the mass of each block is m. Neglect the mass of the pulleys and cord.

A 40-lb suitcase slides from rest 20 ft down the smooth ramp. Determine the point where it strikes the
ground at C. How long does it take to go from A to C? Note: A smooth surface doesn’t have a frictional
force.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________
A 60-kg suitcase slides from rest 5 m down the smooth ramp. Determine the distance R where it strikes
the ground at B. How long does it take to go from A to B?

If the 50-kg crate starts from rest and achieves a velocity of v = 4 m/s when it travels a distance of 5 m to
the right, determine the magnitude of force P acting on the crate. The coefficient of kinetic friction
between the crate and the ground is mk = 0.3.

ACTIVITY 4 What I Know Chart (PART 2)

{*This serves as the student’s review and summary of what was learned from the session. The teacher asks
students to monitor how their knowledge has changed by reviewing the questions in the What I Know Chart
from Activity 1 and write their answers to the questions based on what they now know on the third column of
the chart.}
Now, go back to Activity 1 and answer the questions filling out the “What I Know Column” after going through
the main lesson and answering the questions from Activity 3.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 5 Check For Understanding

{*Teacher creates a short quiz for students to check how well they understood the lesson.
**Teacher adds this prompt for self-assessment: “Check your answers against the Key to Corrections found at
the end of this SAS. Write your score on your paper.”}
Now, I’m sure that you have learned enough and are ready to take a short quiz. Of course, the questions that
will be given are more challenging than Activity 3. The schedule will be announced in class. Instructions:
You will write your solutions and answers in a short coupon bond. You may use more than one paper. But only
a limited number will be submitted. (Note: For problem solving type: for values less than one, round off to up to
four significant figures {e.g. 0.1234, 0.5780, 0.0069 are values with four significant figures}, and for values
greater than one, round off to three decimal places {e.g. 3.178, 222.900, 93.232 are values with three decimal
places}).

C. LESSON WRAP-UP
ACTIVITY 6 Thinking About Learning
{*Teacher directs the student to mark their place in the work tracker which is simply a visual to help students
track how much work they have accomplished and how much work there is left to do. This tracker will be part
of the Student’s Activity Sheet.
**To develop habits on Thinking About Learning, teacher writes a question or two that may ask students about
their learning experience, if they met the learning target, what they found difficult/ easy about the topic or
experience, what strategies worked for them or not, etc.}

You are done with the session! Let’s track your progress!
Period 1 Period 2 Period 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ACTIVITY 6.1 Unanswered Questions

In this activity, I’m asking you to write your questions that are still unanswered at this time of reading. If there’s
any thing that still confuses you, don’t hesitate to write them down below. Make sure though that these
questions are not included in the FAQs section written below/ on the next page. So I suggest you to read the
FAQs first before proceeding to this activity 😊
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 9
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

FREQUENTLY ASKED QUESTIONS (FAQs)

{*Teacher writes 2-3 questions with answers that they anticipate students would ask about the topic. FAQs help
make up for decreased opportunity for students to ask for clarifications or explore related topics.}
QUESTION What is rolling resistance?

ANSWER Rolling resistance or rolling friction or rolling drag is an additional frictional force resisting the
motion of a body such as a wheel, a ball, or a tire rolling on a surface.
QUESTION What will be the sign convention of the equation of motion when you rotate the axes?

ANSWER It’s up to you. You can always assume the direction as long as you should follow that
convention up until the end.

TEACHER-LED ACTIVITIES
{*These are standard instructions for teachers.
A. If this session happens to be a face-to-face, in-classroom learning session:
1) Collect completed work in the SAS.
2) Allocate your contact time with students to individual or small group mentoring, monitoring, and student
consultations.
3) You may administer summative assessments (quizzes, demonstrations, graded recitation,
presentations, performance tasks) during face-to-face sessions.
4) You may also explore supplementary activities that foster collaboration, provided that social distancing
is observed.
5) You may provide supplementary content via videos, etc.
It is important to remember that students who cannot make it to face-to-face, in-classroom sessions for health
and safety reasons, should not be given lower grades for missing in-class activities and should be given
alternative summative tests.
B. If this session happens to be an at-home learning session for the students:
1) Check and grade collected SAS and other input from students.
2) Schedule phone calls/ virtual calls/ virtual chats to individual students or small groups of students to
monitor work, provide guidance, answer questions, and check understanding.}

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
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Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

L E S S O N T I T L E : C U RV I L I N E A R M O T I O N : Materials:
RECTANGULAR COMPONENTS Textbook, Notebook, Pen, Calculator

Lesson Objectives: References:

Main Textbook: Engineering Mechanics: Statics and
At the end of this module, you should be able to: Dynamics by R.C. Hibbeler (https://drive.google.com/
1. Obtain the time derivatives of a particle in a file/d/1LSZHHta3BIh7GugVLaTlyz4oPUGvFqKi/
curvilinear path. view?usp=sharing)
You can copy the links and open them on Google to
obtain a softcopy of the file. You can place it on your
phone for easy access wherever you are 😄

Productivity Tip:

DAYS, MONTHS, YEARS FROM NOW, THE

THINGS YOU ARE NOW STRESSING ABOUT
WON’T EVEN MATTER. SO JUST DO IT. -
Unknown
You may not realize it everyday but remember when you
stressed yourself about the time you failed a quiz in
Statics. Now you’re here! Now you’re taking up
Dynamics! You were sad or even cried during that time
but here you are, learning as much as you can from your
mistakes and celebrating your victories, no matter how
small. 😀

A. LESSON PREVIEW
Introduction
Now, it’s time for solve us to solve problems! In this
module, we will be extracting the curvilinear motion into its
rectangular components. We are going to dissolve the
motion into three axes - the x-axis, y-axis, and z-axis. This
is going to help us simplify the motion. We will be talking
about first time derivatives (velocity) and second time
derivatives (acceleration). Let’s get to it! Turn to the next
page!

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 1 What I Know Chart (PART 1)

Instruction: Before proceeding to the main lesson, write three things you know or have learned from the
previous lesson in the first column “What I Know”. The third column “What I Learned” will be filled out at the
end of the module.

WHAT I KNOW QUESTIONS WHAT I LEARNED (Activity 4)

Does velocity act tangent to the

1 1
path?

Does acceleration act tangent

2 2
to the path?

What is the total magnitude of

3 3 acceleration given the values of
a in three dimensions?

B. MAIN LESSON
ACTIVITY 2 Content Notes
In this next activity, you will be presented with presentation slides and a detailed explanation of the content of
each slide. You are to write down the key points of the lecture that are critical to your evaluation. Make sure
that your pen and paper are beside you so you can go with the flow of the lesson. Enjoy reading and have fun
learning 😍 See next page for the main lesson!

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

Occasionally the motion of a particle can best be described along a path that can be expressed in terms of
its x, y, z coordinates.

POSITION
If the particle is at point (x, y, z) on the curved path s shown below, then its location is defined by the
position vector:

At any instant, the magnitude of r is:

And the direction of r is specified by the unit vector ur = r/r.

VELOCITY
The first time derivative of r yields the velocity of the particle, hence (see next page):

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Dynamics of Rigid Bodies
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Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

When taking this derivative, it is necessary to account for changes in both the magnitude and direction of
each of the vector’s components. For example, the derivative of the i component of r is:

The second term on the right side is zero, provided the x, y, z reference frame is fixed, and therefore the
direction (and the magnitude) of i does not change with time. Differentiation of the j and k components may
be carried out in a similar manner, which yields the final result,

where:

The dot notation represents the first time derivatives of x = x(t), y = y(t), z = z(t), respectively.
The velocity has a magnitude that is found from, and is specified by the unit vector uv = v/v. This always
acts tangent to the path.

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACCELERATION
The acceleration of the particle is given by:

Here ax , ay , az represent, respectively, the first time derivatives of vx = vx(t), vy = vy(t), vz = vz(t), or the
second time derivatives of the functions x = x(t), y = y(t), z = z(t).
The acceleration has a magnitude of, and a direction specified by the unit vector ua = a/a. In general, a will
not be tangent to the path.

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Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 1

At any instant, the horizontal position of the weather balloon is defined by x = (8t) ft, where t is in seconds. If
the equation of the path is y = x2/10, determine the magnitude of the velocity and acceleration when t = 2 s.

Solution:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

SAMPLE PROBLEM NO. 2

For a short time, the path of the plane is described by y = (0.001x2) m. If the plane is rising with a constant
upward velocity of 10 m/s, determine the magnitudes of the velocity and acceleration of the plane when it
reaches an altitude of y = 100 m.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

Solution:

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 3 Skill-Building Activities

Now, you will be presented with a brief activity that contains both objective and problem solving type questions.
You can check your answers against the Key to Corrections found at the end of this module. But as much as
possible, don’t peek if you still haven’t tried answering the questions 😅 Write your score on the space
provided. Good Luck! Have fun answering!

WHAT’S
Activity 3 Questions YOUR
SCORE?

The particle travels along the path defined by the parabola. If the component of velocity along the x axis
is vx = (5t) ft/s, determine the particle’s distance from the origin O and the magnitude of its acceleration
when t = 1 s. When t = 0, x = 0, y = 0.

ACTIVITY 4 What I Know Chart (PART 2)

Now, go back to Activity 1 and answer the questions filling out the “What I Know Column” after going through
the main lesson and answering the questions from Activity 3.

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 BES 026
Dynamics of Rigid Bodies
Students’ Activity Sheet Module # 10
Home Module
!
Name: __________________________________
Section:_________________________________ Class Number: _______________
Schedule: _______________________________ Date: _______________

ACTIVITY 5 Check For Understanding

Now, I’m sure that you have learned enough and are ready to take a short quiz. Of course, the questions that
will be given are more challenging than Activity 3. The schedule will be announced in class. Instructions:
You will write your solutions and answers in a short coupon bond. You may use more than one paper. But only
a limited number will be submitted. (Note: For problem solving type: for values less than one, round off to up to
four significant figures {e.g. 0.1234, 0.5780, 0.0069 are values with four significant figures}, and for values
greater than one, round off to three decimal places {e.g. 3.178, 222.900, 93.232 are values with three decimal
places}).

C. LESSON WRAP-UP
ACTIVITY 6 Thinking About Learning

You are done with the session! Let’s track your progress!
Period 1 Period 2 Period 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ACTIVITY 6.1 Unanswered Questions

In this activity, I’m asking you to write your questions that are still unanswered at this time of reading. If there’s
any thing that still confuses you, don’t hesitate to write them down below. Make sure though that these
questions are not included in the FAQs section written below/ on the next page. So I suggest you to read the
FAQs first before proceeding to this activity 😊
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________

FREQUENTLY ASKED QUESTION (FAQ)

QUESTION What does the two dot notation mean?

ANSWER The dot notation represents the first time derivatives of x = x(t), y = y(t), z = z(t), respectively
and so the two dot notation represents the second time derivatives of x = x(t), y = y(t), z = z(t),
respectively.

KEY TO CORRECTIONS
See Answer Key from ACTIVITY 3 Skill-Building Activities
1. d = 4.00 ft (10 points), a = 37.8 ft/s2 (10 points).

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